Cross-layer optimization of wireless multi-hop networks with network coding

Abstract

In this paper we present a cross-layer algorithm for joint optimization of congestion control, routing, and scheduling in wireless multi-hop networks with network coding. We introduce virtual flow variables in the formulation of capacity region of the networks. The utility maximization problem subject to constraints on the capacity region is solved using dual decomposition and subgradient method, based on which a new queuing model is obtained and which also can result in a cross-layer algorithm for distributed implementation. The new queuing model can facilitate coding operations and reduce coding complexity. Simulation results show that network coding in the proposed joint optimization algorithm can interact adaptively and optimally with other components in different layers, and thus yield higher performance than the routing scheme without network coding.

Full text

Page 1

Cross-layer Optimization of Wireless Multi-hop Networks
with Network Coding
Lei You1,2 Ping Wu1 Zhiwen Pan3 Honglin Hu4 Junde Song2 Mei Song2
1Signals & Systems, Dept. of Engineering Sciences, Uppsala University, Uppsala, Sweden
2PCN&CAD Certer, Beijing University of Posts and Telecommunications, Beijing, China
3National Mobile Communication Research Laboratory, Southeast University, Nanjing, Jiangsu, China
4Shanghai Research Center for Wireless Communications, Shanghai, China
youlei.feng@gmail.com, Ping.Wu@angstrom.uu.se, pzw@seu.edu.cn, Honglin.hu@shrcwc.org, {songm, jdsong}@bupt.edu.cn
Abstract—In this paper we present a cross-layer algorithm for
joint optimization of congestion control, routing, and scheduling
in wireless multi-hop networks with network coding. We
introduce virtual flow variables in the formulation of capacity
region of the networks. The utility maximization problem subject
to constraints on the capacity region is solved using dual
decomposition and subgradient method, based on which a new
queuing model is obtained and which also can result in a cross-
layer algorithm for distributed implementation. The new queuing
model can facilitate coding operations and reduce coding
complexity. Simulation results show that network coding in the
proposed joint optimization algorithm can interact adaptively
and optimally with other components in different layers, and thus
yield higher performance than the routing scheme without
network coding.
Keywords-Wireless multi-hop network; Network coding; Cross-
layer Optimization
I.
INTRODUCTION
Network coding (NC) has emerged as a promising
mechanism to efficiently utilize the resource of both wired and
wireless networks. The pioneering work by Ahlswede et al. [1]
has showed that maximum capacity of a multicast session can
be achieved with NC. It is more attractive to implement NC in
wireless networks due to the broadcast property of wireless
medium, i.e., a single transmission from a node can be
received by all its neighbours. A typical wireless NC is the
opportunistic XOR coding, e.g., COPE scheme proposed in [2]
that can identify coding opportunities and forward multiple
packets (coded together) in a single transmission using
broadcast advantage. However, integration of NC into the
existing architecture of wireless network is not straightforward.
Network coding can not work as an independent function in a
specific layer. Optimal control of wireless multi-hop networks
with NC involves interaction of NC with all the other
functions in different layers (e.g., congestion control in
transport layer, routing in network layer and scheduling in
MAC layer).
Cross-layer optimization of wireless communication
networks has been a very active research area in recent years.
Backpressure-based approaches that determine routing and
scheduling using queue backlog information is widely used for
cross-layer design due to its optimality [14]. Joint optimization
of congestion control, routing and scheduling for wireless
multi-hop networks has been extensively studied using dual
decomposition and sub-gradient method [3] [4]. Backpressure-
based control algorithms for joint routing, scheduling and
network coding across multiple uncast sessions are developed
in [5] and [6] based on the poison-remedy approach [13]. The
problem of coding-aware routing for wireless multi-hop
networks is solved using linear programming, which is a
centralized solution [7]. A backpressure-based optimal policy
for joint scheduling and network coding with predetermined
routing is proposed [8]. An oblivious backpressure algorithm
for joint routing, scheduling and network coding is proposed
in [9] for energy efficiency of wireless networks. However,
optimal coding operation involves searching all the packets in
all the queues, which results in a high complexity. In this
paper, we consider joint optimization of congestion control,
routing, scheduling and network coding for a wireless multi-
hop network.
The paper is organized in the following way. In Section II
the system model is discussed and the capacity region is
formulated by introducing virtual flow variables. Then in
Section III, the utility maximization problem is solved using
dual decomposition and subgradient method, a new queuing
model is obtained from the dual variables and then a cross-
layer optimization algorithm is proposed. The simulation
results are presented in Section IV, and the paper is concluded
by Section V.
II.
SYSTEM MODEL AND CAPACITY REGION WITH
NETWORK CODING
A. System model
Consider a wireless multi-hop network, whose topology is
represented by a directed graph
and a link set L. Link between nodes i and j is denoted by
. The maximum transmission rate of each link
fixed value,
. We assume that time is slotted and all the
nodes in the network are synchronized.
with a node set N
has a
( , )
N LG

( , )
i j
( , )
i j
ijc

Fig.1 OTIC network coding
ix as the data rate of the session generated at node i
and destined to node d. We consider a multi-commodity flow
model and view packets to the same destination as one
commodity. Let
{ :1,...,
Dd d

It is easy to verify that it is inefficient to apply network coding
to the sessions to a common destination. Here we consider
one-hop network coding between local two-way commodities
at intermediate nodes. Specifically, as showed in Fig. 1,
assuming a commodity d traversing links ( , )
Denote
d
}
N
be the set of commodities.
j i and ( , )
i k , and
This work is sponsored by VINNOVA Sweden, CSC China, and the
Ministry of Science and Technology China through National International
Science and Technology Cooperation Project (NO. 2008DFA12090).

Page 2

another commodity d traversing links ( , )
node i XORs the packets from d and d and sends the coded
packet through one broadcast to nodes j and k, where the
coded packets will be decoded immediately using the packets
that had traversed the node before. We refer to such a coding
scheme as One-hop Two-way Inter-Commodity (OTIC)
network coding.
We assume that each node uses a single half-duplex
transceiver so that the node can not receive or transmit
simultaneously. For OTIC network coding, a coded packet
contains two packets that are from two different commodities
(d and d’) sent in reverse directions. Thus, there are two kinds
of transmission: unicast transmission of native packets on a
single link and broadcast transmission of coded packets on
two links with a common transmitting node and two different
receiving nodes. Denote ( ,
transmitting node i and receiving node set B which contains
two nodes in the OTIC case, i.e., B={ ,
following, we also use ( ,
to denote broadcast link ( ,
Transmission rate is denoted as c
. Let ( ,
transmission on link ( ,
and received by node
))
k
( ,
ij
we have cc
. Let E denote the set of all the
possible broadcast links with each element denoted by e.
We consider K-hop interference model in which links within
a K-hop range can not be active simultaneously. Using the
method in [7], we can construct a conflict graph to identify all
possible independent sets of links. An independent set is such
a set of unicast and broadcast links that can be active
simultaneously, because each is out of the K-hop range of
other. Let denote the set of all the independent sets, and

an independent set that is a combined rate vector (
a
||||
LE

-dimension. The rate vector
dimension, and its l-th element can be written as
if unicas
( )
0 otherwi
se

and the rate vector
element can be expressed in the following manner
if broadca link
( )
0 otherw
ise

The feasible transmission rate region in the link layer using
a time-sharing scheme is the convex hull of all the |
dimension rate vectors in , which can be expressed as


 


k i and ( , )
i j , then
as a broadcast link with
and ||
. In the
i B
.
, which is defined as
denote the broadcast
)
)
( , j k
i B
, ) k
ij
}
j k
2

B
(
ij
)
,(
))
,
ij
)
k
,( , )
j k
min( ,)
iijik
cc

c
j . Thus,
,( , )
j k ijik,(

, )
j k
,(
i

, )
j k
c

,)
uB
rr
 with
ur is of ||
L
-
l
u
cl
r l

t link




Br is of -dimension and its e-th
||
E
e
B
ce
re

st




|||
LE

-
( )( , )
u
r r
)(0,1
Bu
Cor








:( ,
r r
uB
, r

B

),




B. Cross-layer problem formulation
Let
denote the native flow rate of commodity
unicast link , and
commodity
that is coded with commodity d at node i,
broadcasted on link and received by node j. Note that
and are virtual information rates for
transmission on link ( ,
. To decode the coded packets
, ))
k
(
ij
d
ijf
on
the coding flow rate of
correctly after the broadcast transmission, the previous node of
packets of commodity
coded in virtual flow
d
while that of commodity d in flow


as the set of all feasible
commodity pairs. To develop a cross-layer algorithm with a
low coding complexity, we formulate the capacity region of
wireless multi-hop network with network coding by
introducing virtual flow rate variables {
represents the total unicast rates (without coding) of packets at
node i of commodity d that come from node j. With those
virtual flows defined, the capacity region for a wireless multi-
hop network with OTIC NC can be formulated as follows,
,()
,( , )
0,0 ,
ijijj k
ff

,( , )
j k
f
i j k


ji
, )
k

g
 
d
( , )
i j
d
,( )
,( ,
j k
dd
ik
f

,(
,(
ijf
))
k
)
, )
j k
d d


( , (
ij,
)
,()
,( , )
j k
d d
ij
f

,()
,( , )
j k
d d
ij
f
 is node k,
is node j. Denote
,( )
,( , )
j k
dd
ik
f

{( ,
d d
):,,
Pd dD
}
dd
}
d
ji
g
, where
d
ji
g
dd d
,(
,(
( , )
k j
k i
, ( ,)
d
ji
d
ji i
Ld dP
ff


,()
, ,
 
d d
dj


,( )
,( , )
j k
N d

,
d d
ij
f

, ( ,)
,,
d
L
j

d dP
D i





0
,d
f
ik
d

d
f
dj
)
D


,(
,( , )
j k
)
, ( , )
j i
d
d

(1)
( , )
i k
k
dd
iN dL
 
,
(2)
( , )
j i
g
( , )
i j
i

d
i
d
ji
LL
,
d
ij
xgfiN dD i




 

( , )
j i
g

,
d
ji
d
i
L

(3)
,
V

N dD


(4)
0,
d
ji
d
di
，
d
jd
0,,
gg


( , )
i j
d
ijij
d D

r
where
i
 j

N

d

D

(5)

Bj
fr

,
,(
B
)
d d
,
ij
,, (
i B
r
( ,
d d )
, where
, are fixed and finite constants.
P
, ),
fi

 
B (6)
, ir
( ,
u
r r
) ( )

B
,
Co
(
uij
rr ),(
Br
)
B

,
d
iViN d

(7)
D
 
Maximi
s.t. (1)-

ze
(7)
Constraints in (1) present that all the flow variables are
nonnegative, all the flows can not be sent to other nodes by
their destinations, and the packets of the same commodity can
not be coded together; Constraints in (2) and (3) are the flow
conservation law for each commodity at nodes that are not its
destination. (6) defines the constraints on link capacity
allocation. (7) is the constraint on link rate region of the
wireless multi-hop network with OTIC network coding with
time-sharing; (4) represents the constraints on the virtual flow
rates of each commodity at each node. Since the virtual flow
variables are added artificially, each
sufficiently large (e.g., larger than the sum of the input rates of
the corresponding node) so that the capacity region is not
smaller than that without virtual flows.
The objective of our cross-layer optimization problem is to
achieve fairness among all the sessions by maximizing the
sum of utilities of all the sessions subject to the constraints
(1)-(7) on the capacity region as follow,
U

should be set to be
d
iV
)
, i d N

(
d
i
d
i
x
(P1)
where the utility function
sending rate
continuously differentiable function. Maximizing the total
utility can achieve proportional fairness if
Primal problem P1 is a convex problem with linear
constraints and a concave objective function [10]. It can be
solved using interior-point method. But this requires
associated with session i with
(
i
U x )
i
ix is assumed to be a concave, non-decreasing and
( )log(z)
U z 
.

Page 3

centralized computation. We will show that one can use dual
decomposition and subgradient method [3] [4] to P1 to find a
cross-layer optimization solution
implemented in a distributed manner, and to obtain a new
queuing model that can facilitate coding operation as well.
which is possibly
III. CROSS-LAYER OPTIMIZATION ALGORITHM
Since P1 is convex, strong duality holds. Thus, it can be
solved optimally by solving its dual problem. By introducing
Lagrange multipliers {}
ji
p
and {
and (3), respectively, the corresponding partial Lagrangian of
P1 can be written as follows
()
L


   
d
to relax constraints in (2)
}
d
iq
,
,(
,( , )
j k
) ,(
,( , )
ji i k
)
( , )
j i
( , )
i k
k
x
,( ,
d d

)( , )
k j
k i
, ( ,
d d
)
( , )
i j
( , )
j i
()
[]
[]
d
i
d
i
i d N

d
ji
d
ji
d d
ik
d
ji
d d
i N


L d D

LPLP
j

d
i
dd
i
d
jiij
i N d D

where
of session rates, native flow rates, coding flow rates and
virtual flow rates, respectively; p and represent the vectors
of corresponding dual variables. With this Langrangian we can
define the dual objective function as
max(
()
. . (1) and (4)-(7)
st
 
Thus, the dual problem of problem P1 can be written as:
LL
Ux
pgfff
qfg





and g represent the vectors of primal variables



 

nc
x,f,f , g;p,q
nc
x,f, f
q
)
L
D


nc
nc
x,f,f, g
x,f,f , g;p, q
p,q
(8)
Minimize
. .
s t
p
()
00
D

p, q
, q
(D1)
which is also a convex problem [10].
A. Subgradient method to solve D1
Subgradient method is effective to solve dual problem D1
since its objective function (8) is not differentiable [11].
Supposing primal variables
solution of maximization in dual objective function (8) at
point
()
p,q , the subgradients of function (8) at
be expressed as follows, respectively,
,()
,( , )
( , ),( ,)
i kLd dP
kj
Grfxg


Subgradient method finds the optimal solution of D1 by
updating dual variables in each iteration step t using
subgradients in (9) in the following way until the solution
converges.










where ( ) t

is a positive step-size at step and  
Subgradient method converges to the optimum of D1 if
( ) t

is designed appropriately according to the rules in [11].
Since strong duality holds for P1, the primal variables
and are obtained as a
nc
x,f,f
g
d
ji
p and
d
iq c an

d
ji
,(
,( , )
ji i k
)
( , )
k j
k i
,( ,
d d
)
( , )
i j
( , )
j i
d
ji
d
ji
d d
ik
d
ji
d d
j k
LP
ddd
i
d
jiiij
LL
Gr



gfff






  

(9)
( 1)( )( ) (
t
)
( 1)( )( ) (
t
)
d
ji
d
ji
d
i
d
i
d
i
p t p tGr
q tq tGr





 (10)


.
t
max( ,0)
yy

(
variables
) related to the corresponding optimal dual
are a globally optimal solution to P1 [10].
q )
***
nc
*
x ,f ,f , g
*
(p ,
()(
D
where
D
D
*
)

B. Calculating subgradient by dual decomposition
To calculate subgradients (9) in each step t , we need to
determine primal variables
of maximization problem in (8) at point ( (
maximization problem in (8) can be decomposed into the
following three subproblems:
and , which is the solution
g
p
. The
nc
x,f,f
)( ))
tt
,q
123
()()
DD

p,qqp,qp,q
1
0
,
( )
q
max

x
i d
[
N


()]
d
i
d
i
d
i
d
i
Ux

2
ma
()
q x

(11)
( , )
j i
()
4) (5)
  
   
d
ji
d
ji
d
i
x
. . (
st
i N

L d D

gpq


D



  
g
(1),
i N

i N

p,q
3
max
f,f
()
. .
D
st






 
(12)
( , )
i j
,(
j k
)
,( , )
( , )
i j
( , )
L i k
k
( ,)
()
[()( )]
(6) and (7)
dd
i
d
ij ij
L d D

d dd
ji
d
ik
d
ki
d
iji
L d dP
j
fqp
fpppp























nc
p, q
(13)
Subproblems (11), (12) and (13) can be solved separately
and the solutions actually result in a cross-layer optimization
algorithm for joint congestion control, routing, scheduling and
network coding, to be showed in Section III.D. Next, we first
present the queuing model implied by dual variables and q .
p
C. Queuing model and evolution of the queues
If the step-size for updating dual variables is constant in
each step, the variables are updated in a similar manner to the
corresponding queues [3] [4]. For a sufficiently small,
constant step-size, subgradient method can converge to a small
neighbourhood of the optimal solution [11][4]. Similarly,
assuming a constant step-size in (10), i.e.,
associate each of the dual variables in
which results in a new queuing model at each node. Different
from previous work [3] [4] where no network coding was
considered and each node maintains only one queue for each
commodity, each node in the present case maintains several
queues for each commodity d, including single unicast queue,
, buffering packets to be sent out as native packets, and
receiving queues
,
ji
P
( , )
j i

used to buffer all the packets from neighbouring node j. There
is a virtual link between and between each queue
respectively. The packets of commodity d that comes from
neighbour j but are to be sent out as native packets (without
coding) are transferred from
ji
P to
with virtual flow rate
ji
g . Thus, the evolution of the queues in
this queuing model is

















( ) t
with a queue,
(p, )
q


, we can
d
i Q
d
L
, where each queue
d
ji
P is
d
i Q
d
ji
P ,
d
through the virtual link
d
i Q
d
,(
,( , )
j k
) ,(
,( , )
ji i k
)
( , )
i k
, ( ,)( , )
k j
k i
, ( , )
d d
( , )
i j
( , )
j i
()
(1)( )()
d
ji
d d
ik
d
ji
d d
L d dPLP
k j

d
i
d
i
dd
i
d
jiij
LL
gfff
Q tQ tfxg
(1)
d
ji
d
ji
P t
( )
P t
















  



(15)

Page 4

Comparing (15) with (10) for positive constant ( ) t
can find the relationships of queue lengths with corresponding
dual variables,
( )( )/
jiji
P t p t 
and


, we
dd
( )( )/
d
i
d
i
Q tq t 

.
D. Cross-layer optimization algorithm
With the above queuing model, we can use queue length
information instead of dual variables in the dual
decomposition and subgradient method.
Algorithm 1: Cross-layer optimization algorithm for
wireless multi-hop network with OTIC NC
Assuming that the lengths of unicast and receiving queues are
and P
the following operations at slot t:
1) Congestion control: Each session
using the length of local unicast queue
max(0, U [Q ( )/ ])
iii
xt 

, where
function of derivative of utility function U ( )
predetermined constant parameter (step-size).
2) Unicast queue loading: Each node i
 puts the packets generated from session
queue
 finds
( , )
j iL

etwork performs
( )
d
i Q t
( )
d
jit ,
, , ( , )
j iiN dDL
  , the n
d
ix calculates its rate
( )
U( )
d
i ,  is the
d
i Q t at node i as
-1-1
ddd

d
i  is the inverse
( )
d
ix t into unicast
d
i Q , and
argmax(
j

( ) ( ))
d
ji
d
i
P t Q t

ifand
( )
t
0otherwise
P t to queue
d
i
d
ji
j i
Vjj
g




 



for each commodity d, then
sets
d
i

and transfers
packets from queue
3) Link Scheduling: Each node i
 finds
d
argmax(
d D

( )
d
ji
( )
d
i Q t at rate
( )
t .
d
ji
( )
t
( )0
d
PQ t

g
( )( ))
d
ij
Q t
i
P t

and then calculates a weight
t for each unicast link (i, j);
( ))
d


max(0,( )
t
d
i
d
ijij
wQP



 finds

and then
calculates
broadcast link (i, ( j, k)).
 Some scheduling algorithm picks an optimal link
independent set  that satisfies the relation




4) Network coding and Routing: Each node i
 XORs packets from receiving queues
respectively, and broadcasts the coded packets on links ( , )
and ( , )
i k at rate min( ,
ijik
c c
if broadcast link ( , ( , ))) ,
(,)
d d
and
for each
( ,
(
P
) .(
) 0

) 0,

(,) argmax
PP
P

[()( )]
d
ji
d
ik
dd
kiij
d
ji
ddd
ik kiij
d dP
ddPPPP


 



,( , )
j k
( ( )
t
( ))
t
(( )
t
( ))
t
d
ji
ddd
iikkiij
wPPPP



,( , )
j k
( , )( , )( , )
L i k
k
max

( , )( ,( , ))
r i
uijBi
i N i j

Li N i j

L
j
r i j w j k w




    

(16)

d
ji
P

and
d
ki
P
i j



ij k

 are the commodity pair that attains
,( , )
j ki
w
.
 transmits the packet
at rate
commodity pair that attains
s from unicast queue
if unicast link ( , )
on lin
d
i Q

k ( , )
i j
ijc
i j


and d are th
ij
w
5) Queue information exchanging:
At the end of this slot, all the que
e
.

to (15). All the nodes exchange their queue length information
with their neighbours through a certain control channel.
From step 4) in the algorithm, we can see that the developed
queuing model facilitates the optimal coding operation.
Different from one queue per one commodity model, it neither
has to maintain for each packet the set of nodes which it
comes from nor has to search all the queues for packets that
can be coded together in each slot. For network coding, a node
just needs to pick packets from corresponding receiving
queues. Thus the complexity of finding valid coding
opportunity is reduced.
E. Link scheduling algorithm
Note that Algorithm 1 is possible to be implemented in a
distributed way except for link scheduling (16) which is a
maximum weighted independent set problem that is NP-hard
and needs a centralized solution under general K-hop
interference model.
Since in this paper we are mainly interested in the optimal
interaction of functions at different layers with network coding,
we test the proposed cross-layer optimization algorithm using
a centralized Greedy Maximal Scheduling (GMS) algorithm
[12] for link scheduling (16) in the simulation in the next
section. GMS each time schedules link with the highest weight
( or ) in (16) and drops the links in the interfering
range of the selected link until all the links in the network are
either scheduled or dropped. However, the design and
performance analysis of the distributed link scheduling
algorithm with low complexity under more general
interference model are to be done in our future work.
ij
w
,( , )
j ki
w
IV. SIMULATIONS RESULTS
In this section we evaluate the performance of the proposed
cross-layer optimization algorithm through simulations. The
topology and link rates used herein are showed in Fig. 2. Note
that we set
21324264
cccc

There exist two sessions:
from node 9 to node 3. We set the step-size
utility function
( )ln( )
U xx

. We assume primary interference
models in which links share a common transmitting node and
receiving nodes can not be active simultaneously, and we use
the greedy maximal scheduling (GMS) algorithm as stated in
Section III.E for all the simulations.
0
to get an acyclic topology.
9
1x from node 1 to node 9, and
3
9x
0.01
 
and use

Fig.2 Topology used in the simulations.
A. Comparison
Comparison of Algorithm 1 is made in terms of converged
average session rate, with the cross-layer algorithm based on a
pure routing scheme without network coding developed in
[10]. As shown in Fig. 3, we see that from Algorithm 1 the
converged average rates of flows 9-1 and 3-9 are
and , respectively, while from the algorithm
without network coding the converged average rates

9
1
0.9231
x 
3
9
0.8451
x 
ues are updated according

Page 5

V.
CONCLUSION
are
the performance of the cross-layer optimization with NC is
improved by up to 17.88% in this scenario.
and , respectively. This shows that
9
1
0.75074
x 
3
9
0.7497
x 
In this paper, we proposed a cross-layer algorithm for joint
optimization of congestion control, routing, scheduling and
network coding for a wireless multi-hop network with network
coding. Virtual flow variables are introduced to formulate the
capacity region. After solving the utility maximization
problem subject to constraints on the capacity region using
dual decomposition and subgradient method, we obtain a
cross-layer optimization algorithm with a new queuing model
that can reduce the complexity of the coding operation.
Simulations have showed that, with the proposed algorithm
and queuing model, network coding can interact adaptively
and optimally with the other components in the network layer.
REFERENCES
[1] R. Ahlswede, N. Cai, S.-Y. Li, and R. Yeung. Network information flow.
IEEE Trans. Info. Theory, 46(4):1204–1216, Jul. 2000.
[2] S. Katti, H. Rahul, W. Hu, D. Katabi, M. M´edard, and J. Crowcroft,
“XORs in the air: Practical wireless network coding,” in Proc. of ACM
SIGCOMM, Oct. 2006, pp. 243–254.
[3] X. Lin and N. Shroff, Joint rate control and scheduling in multihop
wireless networks, 43th IEEE CDC, pp.1484-1489, Vol.2, 2004..
[4] L. Chen, S. H. Low, M. Chiang, and J. C. Doyle, “Cross-layer
congestion control, routing and scheduling design in ad hoc wireless
networks,” in Proc. of IEEE Infocom, Apr. 2006.
[5] T. Ho, Y. Chang, and K. J. Han, “On constructive network coding for
multiple unicasts,” in Proc. of Allerton Conf. on Comm., Contr. Comput.,
Sept. 2006.
[6] A. Eryilmaz and D. S. Lun, “Control for inter-session network coding,”
in Proc. of Workshop on Network Coding, Theory & Applications, 2007.
[7] S. Sengupta, S. Rayanchu, and S. Banerjee, “An analysis of wireless
network coding for unicast sessions: The case for coding-aware routing,”
in Proc. of IEEE Infocom, May 2007, pp. 1028–1036.
[8] P. Chaporkar, A. Proutiere. Adaptive network coding and scheduling for
maximizing throughput in wireless networks. in Proc. of MobiCom’07.
[9] Tao Cui, Lijun Chen, Ho, T., “ Energy Efficient Opportunistic Network
Coding for Wireless Networks,” in Proc. of IEEE INFOCOM 2008.
[10] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge, U.K.:
Cambridge Univ. Press, 2004.
[11] D. P. Bertsekas, Nonlinear Programming, 2nd ed. Belmont, MA: Athena
Scientific, 1999.
[12] X. Lin and N. Shroff, “The impact of imperfect scheduling on cross-
layer congestion control in wireless networks,” IEEE/ACM Trans.
Networking, vol. 14, no. 2, pp. 302–315, April 2006.
[13] D. Traskov, N. Ratnakar, D. S. Lun, R. Koetter, and M. M´edard,
“Network coding for multiple unicasts: An approach based on linear
optimization,” in Proc. of IEEE ISIT, July 2006, pp. 1758–1762
[14] L. Tassiulas and A. F. Ephremides. , “Stability properties of constrained
queueing systems and scheduling policies for maximum throughput in
multihop radio networks, ” IEEE Transactions on Automatic Control,
Dec.1992,37(12):1936–1948,
2
2000 40006000800010000 120001400016000
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Number of Iterations
Average Rates of Sessions


Session from 1 to 9 --with NC
Session from 9 to 3 --with NC
Session from 1 to 9 --without NC
Session from 9 to 3 --without NC

Fig.3 Converged average rates with NC and without NC
B. Allocation of flow rates on all the links
Tables 1 and 2 show the average flow rate allocations on each
link for cross-layer optimization algorithms without network
coding (in [10]) and with network coding (Algorithm 1),
respectively. From the tables it follows three observations:
(1) Without network coding, flow
>4->6->9 to avoid interference with flow
network coding, part of flow
>5->6->9 to create coding opportunity with
This is because network coding can increase the
utilization of node 6 (which is a bottleneck) by
broadcasting the coded packets. This shows that
Algorithm 1 can find the coding aware multipath routing
in an optimal and adaptive way.
(2) Nodes 5 and 6 take all coding opportunities for the two
flows to increase the utilization of wireless medium
through broadcast advantage. The average coded flow
rates on links (5, 3), (5, 6), (6, 5) and (6, 7) are about 0.54
(if we ignore the imprecision of computation in the
simulation).
(3) Only part of the two flows meeting at nodes 7 and 8 are
coded together. This is because the nodes are less
congested than nodes 5 and 6, and thus it is no use of
applying coding all the time. This can reduce the
complexity of coding and decoding operation.
These observations show that Algorithm 1 can determine
necessary network coding and perform joint congestion
control, routing, scheduling and network coding in an adaptive
and optimal manner.
Table.1. Allocation of average flow rates on each link without network coding
Link
（（1,2））
Flow 9-1 flow rate on each link 0.7507
Link
（（9,8））
Flow 3-9 flow rate on each link 0.7497
9
1x only follows path 1->2-
9x , while with
1x goes along path 1->2->3-
3
9
3
9x at node 6.
.
（（2,4）
0.7507
（（8,7）
0.7497
（（4,6）
0.7507
（（7,6）
0.7497
）（（2,3）
0
（（6,5）
0.7497
）（（3,5）
0
（（5,3）
0.7497
）（（5,6）
0
--------
）（（6,7）
0.7507
-------
））
（（7,8）
0.7507
-------
）
（（8,9）
0.7507
------
）
））））

Table.2. Allocation of average flow rates on each link with network coding
（（1,2））
（（2,4）（）（4,6））
Coded flow rate 0 0 0
Uncoded flow rate 0.9231
0.3866
0.3866
Link
（（9,8））
（（8,7）（）（7,6）
Coded flow rate 0 0.3257 0.5455
Uncoded flow rate 0.8451 0.5215 0.2997
Link
（（2,3）
0
0.5380
（（6,5）
0.5410
0.3042
（（3,5）
0
0.5385
（（5,3）
0.5335
0.3117
）（（5,6）
0.5335
0.0075
--------


）（（6,7）
0.5410
0.3821
----------


））
（（7,8）
0.5455
0.3776
--------


）
（（8,9）
0.3257
0.5974
-------


）
Flow
9-1
）））
Flow
3-9